Question

find the critical points of and use the second derivative test (if possible) to determine whether each corresponds to a local minimum or maximum. let

Answer 1

To find the critical points of a function and determine whether they correspond to a local minimum or maximum, follow these steps: Find the derivative, solve for x when the derivative equals zero, take the second derivative, and substitute the critical points into the second derivative.

Step-by-step explanation:

To find the critical points of a function and determine whether they correspond to a local minimum or maximum, you need to follow these steps:

1. Find the derivative of the function.
2. Solve for x when the derivative equals zero to find the critical points.
3. Take the second derivative of the function.
4. Substitute the critical points into the second derivative.
5. If the second derivative is positive at a critical point, it corresponds to a local minimum. If the second derivative is negative at a critical point, it corresponds to a local maximum.

For example, let's say we have the function f(x) = x^2 - 6x + 8. The derivative is f'(x) = 2x - 6, and the second derivative is f''(x) = 2. Setting the derivative equal to zero, we get 2x - 6 = 0, which gives us x = 3. Substituting x = 3 into the second derivative, we get f''(3) = 2, which is positive. Therefore, x = 3 corresponds to a local minimum.